Chapter 4: Sequences and Series

Formula Name	Expression
General Term of Arithmetic Sequence	$a_{n} = a + (n - 1) d$
Sum of $n$ Terms of Arithmetic Series	$S_{n} = \frac{n}{2} [2 a + (n - 1) d]$ or $S_{n} = \frac{n}{2} (a + l)$
General Term of Geometric Sequence	$a_{n} = a r^{n - 1}$
Sum of $n$ Terms of Geometric Series	$S_{n} = \frac{a (1 - r^{n})}{1 - r}$ for $r \neq 1$
Sum of Infinite Geometric Series	$S = \frac{a}{1 - r}$ for $\| r \| < 1$
Arithmetic Mean	$A = \frac{a + b}{2}$
Geometric Mean	$G = \pm \sqrt{a b}$
Harmonic Mean	$H = \frac{2 a b}{a + b}$
Triangular Sequence	$T_{n} = \frac{n (n + 1)}{2}$
Pascal Sequence	$(\binom{n}{r}) = \frac{n!}{r! (n - r)!}$
Relation Between A.M, G.M, H.M	$G^{2} = A H$
Recurring Decimal to Fraction	$0. \overset{―}{a} = \frac{a}{9}$

✍️ Guess-Paper Questions (Solved)

📝 Exercise 4.1

Q4. Find the values of nth term of triangular sequence when

n = 7, 9, 12

and

16

Q6. Find the Pascal sequence when

n = 5, 6, 7

and

8

Solution:

The Pascal sequence for the $n$ th row is given by $(\binom{n}{r})$ for $r = 0, 1, \dots, n$ .

(i) For $n = 5$ :

$(\binom{5}{0}) = 1$ , $(\binom{5}{1}) = 5$ , $(\binom{5}{2}) = 10$ , $(\binom{5}{3}) = 10$ , $(\binom{5}{4}) = 5$ , $(\binom{5}{5}) = 1$

Sequence: $1, 5, 10, 10, 5, 1$

(ii) For $n = 6$ :

$(\binom{6}{0}) = 1$ , $(\binom{6}{1}) = 6$ , $(\binom{6}{2}) = 15$ , $(\binom{6}{3}) = 20$ , $(\binom{6}{4}) = 15$ , $(\binom{6}{5}) = 6$ , $(\binom{6}{6}) = 1$

Sequence: $1, 6, 15, 20, 15, 6, 1$

(iii) For $n = 7$ :

$(\binom{7}{0}) = 1$ , $(\binom{7}{1}) = 7$ , $(\binom{7}{2}) = 21$ , $(\binom{7}{3}) = 35$ , $(\binom{7}{4}) = 35$ , $(\binom{7}{5}) = 21$ , $(\binom{7}{6}) = 7$ , $(\binom{7}{7}) = 1$

Sequence: $1, 7, 21, 35, 35, 21, 7, 1$

(iv) For $n = 8$ :

$(\binom{8}{0}) = 1$ , $(\binom{8}{1}) = 8$ , $(\binom{8}{2}) = 28$ , $(\binom{8}{3}) = 56$ , $(\binom{8}{4}) = 70$ , $(\binom{8}{5}) = 56$ , $(\binom{8}{6}) = 28$ , $(\binom{8}{7}) = 8$ , $(\binom{8}{8}) = 1$

Sequence: $1, 8, 28, 56, 70, 56, 28, 8, 1$

📝 Exercise 4.2

Q5. Find the 21st term of the A.P if its 6th term is 11 and the 15th term is 47.

Q6. (a) Which term of the sequence

- 15, - 9, - 3, \dots

is 75?
(b) Which term of the sequence

5, 2, - 1, \dots

- 85

?
(c) Which term of the sequence

- 2, 4, 10, \dots

is 148?

Q9. If

\frac{1}{a}, \frac{1}{b}, \frac{1}{c}

are in A.P, show that

b = \frac{2 a c}{a + c}

Q11. An object falling from rest, falls 12 metres during the first second, 24 metres during the next second, 36 metres during the third second and so on. How much will it fall during the 8th second?

Q12. A man deposits Rs. 13,000 in a bank in the first month; Rs. 14,500 in the second month; Rs. 16,000 in third month and so on. Find how much he has to deposit in the bank at the end of a year.

Q13. A boy saves Rs. 200 at the end of the first week and goes on increasing his saving for Rs. 25 weekly. After how many weeks, his weekly saving will be Rs. 2000.

📝 Exercise 4.3

Q1(i). Find the A.M between 18 and 26.

Q2. Insert three A.Ms. between 3 and 11.

Q5. Insert 6 A.Ms. between

- 30

and

30

Q6. If 5 and 8 are two A.Ms. between

c

and

d

, find

c

and

d

Q7. Find

n

so that

\frac{a^{n + 5} + b^{n + 5}}{a^{n + 4} + b^{n + 4}}

may be the A.M. between

a

and

b

📝 Exercise 4.4

Q4. If

S_{n} = n (2 n - 1)

, then find the series.

Q5. Find the sum of first 100 natural numbers which are neither exactly divisible by 3 nor by 7.

Q7. The sum of

n

terms of two arithmetic series are in the ratio

3 n + 2 : n + 1

. Find the ratio of their 8th terms.

Q8. The sum of three numbers in A.P. is 27 and their product is 405. Find the numbers.

Q9. Find five numbers in A.P. whose sum is 25 and the sum of whose squares is 135.

Q10. The sum of Rs. 42,000 is distributed among five persons so that each person after the first receives Rs. 80 less than the preceding person. How much does each person receive?

Q11. A well digging company charges Rs. 1,250 for the first meter, Rs. 1500 for the second meter and Rs. 1,750 for the third meter and so on. What is the depth of a well that costs Rs. 50,000.

Q12. A man borrows Rs. 25,000 and agrees to repay with a total profit of Rs. 10,000 in 10 installments, each installment being less than the preceding by Rs. 200. What should be his first installment?

📝 Exercise 4.5

Q2. In a G.P,

a_{1} = \frac{5}{9}

a_{6} = \frac{15625}{9}

. Find its

n

th term.

Q3. Find the

n

th term of the geometric sequence if:

\frac{a_{5}}{a_{3}} = \frac{4}{9}

and

a_{2} = \frac{4}{9}

Q5. How many terms are in G.P.

\frac{1}{3}, \frac{1}{12}, \frac{1}{48}, \dots, \frac{1}{196608}

Q6. Find three consecutive numbers in G.P. whose sum is 39 and their product is 729.

Q7. The number of bacteria in a culture increased in G.P from 515,000 to 15,45,000 in 7 days. Find the daily rate of increase, assuming the rate of increase to be constant.

Q8. Find the profit on Rs. 1000 for 5 years at 4% per annum compound profit.

📝 Exercise 4.6

Q2. Insert:
(i) Two G.Ms between 3 and 81.
(ii) Three G.Ms between 2 and

\frac{1}{2}

Q3. The A.M. between two numbers is 29 and the geometric mean is 21, find the numbers.

Q4. For what value of

n

, is

\frac{a^{n - 2} + b^{n - 2}}{a^{n - 3} + b^{n - 3}}

the G.M. between

a

and

b

Q5. Show that the

n

th root of the product of

n

geometric means between

x

and

y

is the geometric mean between

x

and

y

Q6. The A.M. of two positive integral numbers exceeds their (positive) G.M by 2 and their sum is 20. Find the numbers.

Q7. The A.M. between two numbers is 5 and their (positive) G.M. is 4. Find the numbers.

📝 Exercise 4.7

Q1(iv). Find the sum:

0.9 + 0.99 + 0.999 + \dots

n

terms.

Q1(v). Find the sum:

7 + 77 + 777 + \dots

n

terms.

Q1(vi). Find the sum:

0.2 + 0.22 + 0.222 + \dots

n

terms.

Q1(vii). Find the sum:

3 + 33 + 333 + \dots

n

terms.

Q2(ii). Find the sum to

n

terms of the series:

r + (1 + k) r^{2} + (1 + k + k^{2}) r^{3} + \dots

, where

k

and

r

are proper fractions.

Q3. If

a_{n} = {(\frac{1}{4})}^{n}

then find its sum up to

n

terms.

Q4. Find the sum of the following infinite geometric series:
(i)

\frac{1}{5} + \frac{1}{25} + \frac{1}{125} + \dots

(ii)

\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots

(iii)

\frac{9}{4} + \frac{3}{2} + 1 + \frac{2}{3} + \dots

(iv)

2 + \sqrt{2} + 1 + \dots

(v)

2 + 1 + 0.5 + \dots

(vi)

4 + 2 \sqrt{2} + 2 + \sqrt{2} + 1 + \dots

(vii)

0.1 + 0.05 + 0.025 + \dots

Q5. Find the Vulgar fraction equivalent to the following recurring decimals:
(i)

0. \dot{7}

(ii)

1.7 \dot{4}

(iii)

0.2 \dot{5} \dot{9}

(iv)

1.1 \dot{4} \dot{8}

(v)

2.2 \dot{3}

Q6. If

a = \frac{b}{2} + \frac{b^{2}}{4} + \frac{b^{3}}{8} + \dots

0 < b < 2

, then prove that

b = \frac{2 a}{1 + a}

Q7. The sum of an infinite geometric series is half the sum of the squares of its terms. If the sum of its first two terms is

4 \frac{1}{2}

, find the series.

Q8. Joining the midpoints of the sides of an equilateral triangle, an equilateral triangle having half the perimeter of the original triangle is obtained. We form a sequence of nested equilateral triangles in this manner with the original triangle having perimeter

\frac{5}{2}

. What will be the total perimeter of all the triangles formed in this way?

📝 Exercise 4.8

Q2. If the 7th and 10th terms of an H.P are

\frac{1}{3}

and

\frac{5}{21}

respectively, find its 14th and 20th terms.

Q3. The first term of H.P is

- \frac{1}{3}

and fifth term is

\frac{1}{5}

. Find its 9th term.

Q4. If the

p

th term of an H.P. is

q

, the

q

th term is

p

: prove that the

(p + q)

th term is

\frac{p q}{(p + q)}

Q5. If the

p

th term of an H.P. is

q

, the

q

th term is

p

: prove that the

(p + q)

th term is

\frac{p q}{(p + q)}

📝 Exercise 4.9

Q1(i). Insert the H.M between

\frac{1}{64}

and

\frac{1}{81}

Q2(i). Insert four H.Ms between

4

and

20

Q4. Prove that the square of the geometric mean of two numbers equals the product of the A.M and the H.M of the two numbers.

Q5. Find

n

so that

\frac{x^{n - 5} + y^{n - 5}}{x^{n - 6} + y^{n - 6}}

may be H.M. between

x

and

y

Q7. Find the arithmetic, harmonic and geometric means between 1 and 9. Also verify that

A H = G^{2}

Q8. The A.M of two numbers is 8 and H.M is 6. Find the numbers.

Q9. The H.M. of two numbers is

\frac{24}{5}

and G.M. is 6. Find the numbers.

📝 Review Exercise 4

Q3. There are

n

A.Ms. between 8 and 32 such that the ratio of the third and 7th means is 3:5, find the value of

n

Q5. How many terms are there in a G.P if

a = 8

a_{n} = \frac{1}{512}

and

r = \frac{1}{2}

Q8. If A.M and H.M between two numbers are 5 and

4 \frac{1}{5}

respectively. Find the numbers.

Q9. If G.M. and H.M. between two numbers are 15 and

5 \frac{10}{13}

respectively. Find the numbers.

Q10. The second term of an H.P is

\frac{1}{2}

and the fifth term is

- \frac{1}{4}

. Find the 12th term.

❓ Multiple Choice Questions (Interactive)

💡 For printing: MCQs will show correct answers automatically

1. A sequence is a function whose domain is set of:

(a) integers

(b) rational numbers

(d) real numbers

2. If H is the harmonic mean between

x

and

y

then H is:

(a)

\frac{2 (x + y)}{x y}

(b)

\frac{x + y}{2 x y}

(c)

\frac{2 x y}{x + y}

(d)

\frac{x y}{x + y}

3. If

a_{n} - a_{n - 1} = n + 1

and

a_{4} = 14

then

a_{5} = -----------

(a) 3

(b) 5

(d) 20

4. A sequence

{a_{n}}

in which

a_{n + 1} - a_{n}

is the same number for all

n \in N

is called:

(a) A.P

(b) G.P

(d) None of these

5. If

a_{n - 1}, a_{n}, a_{n + 1}

are in A.P, then

a_{n}

is called:

(a) A.M

(b) G.M

(d) Mid-point

6. Arithmetic mean between

c

and

d

is:

(a)

\frac{c + d}{2}

(b)

\frac{c + d}{2 c d}

(c)

\frac{2 c d}{c + d}

(d)

\frac{2}{c + d}

7. The harmonic mean between

\sqrt{2}

and

3 \sqrt{2}

is:

(a)

4 \sqrt{2}

(b)

\frac{4}{\sqrt{2}}

(c)

\frac{3}{2} \sqrt{2}

(d) None of these

8. For any G.P the common ratio

r

is equal to:

(a)

\frac{a_{n}}{a_{n + 1}}

(b)

\frac{a_{n - 1}}{a_{n}}

(c)

\frac{a_{n}}{a_{n - 1}}

(d)

a_{n + 1} - a_{n}

for

n \in N, n > 1

9. No term of a G.P is:

(a) 0

(b) 1

(d) Imaginary number

10. The sum of infinite geometric series is a finite number if:

(a)

| r | > 1

(b)

| r | = 1

(c)

| r | \geq 1

(d)

| r | < 1

11. If the reciprocals of the terms of a sequence form an A.P, then it is:

(a) Harmonic sequence

(b) Arithmetic sequence

(d) Series

12. If

\frac{a^{n + 1} + b^{n + 1}}{a^{n} + b^{n}}

is A.M between

a

b

, then

n

is equal to:

(a) 0

(b) -1

(d)

\frac{1}{2}

13. If

1, x - 1, 3

are in A.P., then

x = -----------

(a) 0

(b) 1

(d) 3

14. G.M between

- 2

and 8 is:

(a)

4 i

- 4 i

(b) 4 or

- 4

- 16

(d) 3 or

- 5

15. General term of a sequence is

{(- 1)}^{n} n^{2}

. Its

4^{t h}

term is:

(a) -4

(b) -16

(d) 4

16. If

| r | > 1

then infinite geometric series is \text{-----------}:

(a) Convergent

(b) Divergent

(d) both a and b

17. The harmonic mean of

\frac{1}{3}

and

\frac{2}{5}

is:

(a)

\frac{4}{11}

(b)

\frac{3}{4}

(c)

\frac{5}{11}

(d)

\frac{11}{4}

18. If

\frac{a - b}{b - c} = \frac{a}{b}

, then

a, b

and

c

are in:

(a) A.P ❌

(b) G.P

(d) None of these

🔑 MCQ Answer Key

1. C

2. C

3. D

4. A

5. A

6. A

7. D

8. C

9. A

10. D

11. A

12. A

13. D

14. A

15. B

16. B

17. A

18. B

📚 Core Definitions

🧮 Essential Formulas

✍️ Guess-Paper Questions (Solved)

📝 Exercise 4.1

📝 Exercise 4.2

📝 Exercise 4.3

📝 Exercise 4.4

📝 Exercise 4.5

📝 Exercise 4.6

📝 Exercise 4.7

📝 Exercise 4.8

📝 Exercise 4.9

📝 Review Exercise 4

❓ Multiple Choice Questions (Interactive)

🔑 MCQ Answer Key